Optimal. Leaf size=68 \[ 2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.0278444, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {662, 620, 206} \[ 2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 662
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^4} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+c \int \frac{\sqrt{b x+c x^2}}{x^2} \, dx\\ &=-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+c^2 \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3}+2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0135384, size = 48, normalized size = 0.71 \[ -\frac{2 b \sqrt{x (b+c x)} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x}{b}\right )}{3 x^2 \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 149, normalized size = 2.2 \begin{align*} -{\frac{2}{3\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{4\,c}{3\,{b}^{2}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{16\,{c}^{2}}{3\,{b}^{3}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{c}^{3}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{c}^{3}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{{c}^{2}\sqrt{c{x}^{2}+bx}}{b}}+{c}^{{\frac{3}{2}}}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0361, size = 284, normalized size = 4.18 \begin{align*} \left [\frac{3 \, c^{\frac{3}{2}} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (4 \, c x + b\right )}}{3 \, x^{2}}, -\frac{2 \,{\left (3 \, \sqrt{-c} c x^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (4 \, c x + b\right )}\right )}}{3 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34296, size = 155, normalized size = 2.28 \begin{align*} -c^{\frac{3}{2}} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + b^{3}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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